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Computational Solid Mechanics at Johns Hopkins University

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Johns Hopkins University
Stochastic Multi-Scale Models for Mechanics of Materials
Lori Graham-Brady

November 2015


As part of research programs for NSF and Army Research Labs, Professor Graham-Brady's group has been studying the effect of randomly occurring flaws on the strain-rate dependent strength of brittle materials, such as those used in ceramic armor or in cementitious materials. Because the these materials exhibit a high degree of scatter, a probabilistic framework is very useful in this context. The solution calls for a micromechanics-based study of dynamic crack growth from individual flaws, coupled to a larger macro-scale model of the over-all constitutive properties. Furthermore, we observe that in real materials, the mesoscale distribution of flaw sizes and flaw density varies from one location to the next.


All real structures exhibit behavioral uncertainties due to the inherent randomness in parameters such as the material properties, loading, or geometry. A better understanding of the effects of this randomness on structural performance is central to describing more accurately the structure's reliability, which is critical to developing more consistent and cost-effective design codes. In the context of civil structures, this is the domain of stochastic mechanics. The specific focus of stochastic mechanics research conducted by Professor Graham-Brady's group is in characterization of random structural parameters that lead to random structural response. Current studies apply multi-scale models to incorporate the microstructural sources of randomness in a structure's material properties into a macro-scale quantification of the structure's inherent variability. This physically consistent approach to parameterizing micro-scale variability in mechanical properties provides a much greater degree of confidence in the subsequent stochastic structural models that rely on these parameters to predict reliability of a structure.


The results show that Weibull models of failure do not always hold, in particular for dynamically loaded brittle materials. Under static loading, the most severe material flaws provide the initiation point for failure cracks that travel through the material. Under dynamic loading, the cracks have less time to develop under the rapidly increasing loads and therefore a large number of flaws are activated. This violates one of the primary assumptions of the Weibull model, which assumes that failure is associated only with weakest part of the material. The results also show that as we demand more spatial fidelity in our models in order to capture properly the mechanisms of failure, we also need to include local variability in the model in order to achieve a more accurate and numerically stable model.  


Accurate computational models of dynamic material failure have proven very difficult. In situations where the failure is associated with localizations, such as shear bands and crack tips, homogenized material models typically exhibit significant mesh dependencies. These mesh dependencies require the analyst to tune the model to provide output that correlates with known results; however, such a model falls short of being truly predictive of the behavior of other material structures. The stochastic multi-scale model promises to provide a more physically based and accurate approach to models of dynamic failure. 

Selected Publications

1. Graham-Brady, L. (2010). "Statistical characterization
of meso-scale uniaxial compressive strength in brittle
materials with randomly occurring flaws," International
Journal of Solids and Structures, 47 (18-19): 2398-2413.
2. Daphalapurkar, N.P., Ramesh, K.T., Graham-Brady,
L.L., Molinari, J.F. (2010). "Predicting variability in
the dynamic failure strength of brittle materials
considering preexisting flaws," in press, Journal of the
Mechanics & Physics of Solids.
3. Acton, K. & Graham-Brady, L. (2010). "Elastoplastic
mesoscale homogenization of composite
materials," Journal of Engineering Mechanics, ASCE,
136(5): 613-624.
4. Tootkaboni, M., & Graham-Brady, L. (2010). "A
Multi-scale Spectral Stochastic Method for
Homogenization of Multi-Phase Periodic
Composites with Random Material Properties,"
International Journal of Numerical Methods in
Engineering. 83(1): 59-90.

Core Competencies

  • Stochastic multi-scale models of materials with randomly varying microstructure
  • Using microstructural images to quantify microscale characteristics such as flaw size and shape distributions
  • Computational models of dynamic failure of brittle materials

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Figure: The approach is to calculate a suite of mechanical properties, reflecting local variations in microstructure, rather than a single set of properties

Current Research Team Members:
• Lori Graham-Brady (PI)
• Cynthia Zingale (Ph.D. Candidate)
• Junwei Liu (Ph.D. Candidate)
• Andrew Gaynor (Ph.D. Candidate)
• Seth Tibbitts (M.S.E. Candidate)

Recent Graduates and Co-workers:
• Katherine Acton (Ph.D. 2009) Assistant
Professor, University of Minnesota at
• Mazdak Tootkaboni (Ph.D. 2008) Assistant
Professor, University of Massachusetts at

Current Research Collaborations:
• Kimberly Kurtis (Civil Engineering,
Georgia Tech) Modeling and
experimentation studying strain-rate
dependent mechanical properties of cement
and concrete
• K.T. Ramesh (Mechanical Engineering,
Johns Hopkins University) Modeling
mechanisms of dynamic failure in ceramic